To solve this triangle problem, let's determine the length $b$, which represents the hypotenuse of the triangle. The given triangle includes the following details:

1. The triangle has:
   • One angle of $30^\circ$,
   • One adjacent side of $8 \, \text{cm}$,
   • One opposite side of $7 \, \text{cm}$.

We will use trigonometric relationships to calculate the hypotenuse $b$.

Step-by-Step Solution

Step 1: Use the cosine rule

The hypotenuse $b$ can be found using the formula for cosine: $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$ Substituting $\theta = 30^\circ$ and adjacent side $8$: $\cos(30^\circ) = \frac{8}{b}$

Step 2: Solve for $b$

$b = \frac{8}{\cos(30^\circ)}$

We know $\cos(30^\circ) = \frac{\sqrt{3}}{2} \approx 0.866$, so: $b = \frac{8}{0.866} \approx 9.2 \, \text{cm}$

Thus, the length of $b$ is approximately 9.2 cm (to 1 decimal place).

Final Answer:

$\boxed{b = 9.2 \, \text{cm}}$

Would you like me to explain how to verify this or expand on another part of the solution?

Relative Questions:

1. How do you determine the hypotenuse if you know two sides instead of an angle?
2. What is the significance of the cosine rule in solving non-right triangles?
3. How do you calculate the sine of $30^\circ$, and when would you use it here?
4. How can the Pythagorean theorem assist in solving similar triangle problems?
5. What other methods can be used to construct such a triangle with precision?

Tip:

Always ensure your calculator is set to degrees (not radians) when solving trigonometric problems involving angles in degrees!